We will review Levy stochastic integrals and Ito’s formula.

Let be a semimartingale, where is a martingale and is a process of bounded variation. The problem of stochastic integration is to make sense of objects of the form

If is a Levy process, then there exists , a Brownian motion with covariance matrix in , and an independent Poisson random measure on with Levy measure , such that for each , Levy-Ito decomposition has the form of

Given , define

Let (resp. ) be the linear space of all equivalence classes of mappings which coincide almost everywhere with respect to , and satisfies the following conditions:

- is predictable, (see definition here)
- (resp. )

(resp. ) is the space of maps , which is predictable and (resp. ). Note that and are Hilbert spaces, and and

In this below, we take . We say that an -valued stochastic process is a Levy-type stochastic integral if it can be written in the following form for each ,

where for each , , , , and is predictable. is an -dimensional standard Brownian motion and is an independent Poisson random measure on with compensator and intensity measure , which we will assume is a Levy measure.

Equivalent notation for Levy-type stochastic integrals of (2) is either

or (to emphasize the domains of integration)

Clearly, is semimartingale, but may not be a Levy process.

Consider a Levy-type stochastic integral of the form (3). Let be the continuous part of defined by

Theorem 1 (Ito’s theorem)For each , , with probability 1,

Another representation of above Ito formula is that, if is of (3), then for each , with probability 1, we have

Note that a special case of Ito’s formula yields the following classical chain rule for differentiable functions , when the process is of finite variation:

Here is famous Levy characterization of Brownian motion.

Theorem 2 (Levy’s characterization)Let be a continuous centered martingale, which is adapted to a given filtration . If for each where is a positive definite symmetric matrix, then is an -adapted Brownian motion with covariance .

*Proof:* Fix and define process , then by Ito’s formula, we obtain

By taking integral, and expectation

Hence, .

Now, We define the quadratic variation to the more general case of Levy-type stochastic integrals of the form (3). For each , we define a matrix-valued adapted process by the following prescription for its th entry

Each is almost surely finite, and we have

The term , which sometimes called an Ito correction, arises as a result of the following formal product relations between differentials:

Theorem 3 (Ito’s product formula)If and are real-valued Levy-type stochastic integrals of the form (3). Then for all , we have that

For completeness, we will give another characterization of quadratic variation which is sometimes quite useful. If and are real-valued Levy-type stochastic integrals of the form (3), then for each , w.p.1, we have

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